The Leading Ideal of a Complete Intersection of Height Two, Part III

Abstract

Let (S, 𝔫) be an s-dimensional regular local ring with s > 2, and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. As in [2, 3], we examine the leading form ideal I* of I in the associated graded ring G: = gr_𝔫(S). Let μ _G (I*) = n ≥ 3, and let {ξ₁, ξ₂,…, ξ _n } be a minimal homogeneous system of generators of I* such that ξ₁ = f* and ξ₂ = g*, and c _i : = deg ξ _i  ≤ deg ξ _i+1: = c _i+1 for each i ≤ n − 1. For m ≤ n, we say that K _m : = (ξ₁,…, ξ _m )G is an ideal generated by part of a minimal homogeneous generating set of I*. Let D _i : = GCD(ξ₁,…, ξ _i ) and d _i  = deg D _i for i with 1 ≤ i ≤ m. Let K _m be perfect with ht _G K _m  = 2. We prove that the following are equivalent:

1.	deg ξ i+1 = deg ξ i  + (d i−1 − d i ) +1, for all i with 3 ≤ i ≤ m − 1;
2.	deg ξ i+1 ≤ deg ξ i  + (d i−1 − d i ) +1, for all i with 3 ≤ i ≤ m − 1.

Furthermore, if these equivalent conditions hold, then K _m  = I*. Moreover, if e(G/K _m ) = e(G/I*), we prove that K _m  = I*. We illustrate with several examples in the cases where K _m is or is not perfect.

Key Words:

• Associated graded ring
• Hilbert series
• Ideal of initial forms
• Minimal generating set
• Multiplicity
• Regular local ring

2010 Mathematics Subject Classification:

• Primary 13A30, 13C05
• Secondary 13E05, 13H15

ACKNOWLEDGMENT

Shiro Goto is supported by the Grant-in-Aid for Scientific Researches in Japan (C(2), No. 1364044).

Notes

Communicated by I. Swanson.